Theorem pwnss 3933

Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
pwnss	|- ( A e. V -> -. ~P A C_ A )

Proof of Theorem pwnss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq12 2143	. . . . . . 7 |- ( ( y = { x e. A | x e/ x } /\ y = { x e. A | x e/ x } ) -> ( y e. y <-> { x e. A | x e/ x } e. { x e. A | x e/ x } ) )
2	1	anidms 389	. . . . . 6 |- ( y = { x e. A | x e/ x } -> ( y e. y <-> { x e. A | x e/ x } e. { x e. A | x e/ x } ) )
3	2	notbid 624	. . . . 5 |- ( y = { x e. A | x e/ x } -> ( -. y e. y <-> -. { x e. A | x e/ x } e. { x e. A | x e/ x } ) )
4		df-nel 2340	. . . . . . 7 |- ( x e/ x <-> -. x e. x )
5		eleq12 2143	. . . . . . . . 9 |- ( ( x = y /\ x = y ) -> ( x e. x <-> y e. y ) )
6	5	anidms 389	. . . . . . . 8 |- ( x = y -> ( x e. x <-> y e. y ) )
7	6	notbid 624	. . . . . . 7 |- ( x = y -> ( -. x e. x <-> -. y e. y ) )
8	4, 7	syl5bb 190	. . . . . 6 |- ( x = y -> ( x e/ x <-> -. y e. y ) )
9	8	cbvrabv 2600	. . . . 5 |- { x e. A | x e/ x } = { y e. A | -. y e. y }
10	3, 9	elrab2 2751	. . . 4 |- ( { x e. A | x e/ x } e. { x e. A | x e/ x } <-> ( { x e. A | x e/ x } e. A /\ -. { x e. A | x e/ x } e. { x e. A | x e/ x } ) )
11		pclem6 1305	. . . 4 |- ( ( { x e. A | x e/ x } e. { x e. A | x e/ x } <-> ( { x e. A | x e/ x } e. A /\ -. { x e. A | x e/ x } e. { x e. A | x e/ x } ) ) -> -. { x e. A | x e/ x } e. A )
12	10, 11	ax-mp 7	. . 3 |- -. { x e. A | x e/ x } e. A
13		ssel 2993	. . 3 |- ( ~P A C_ A -> ( { x e. A | x e/ x } e. ~P A -> { x e. A | x e/ x } e. A ) )
14	12, 13	mtoi 622	. 2 |- ( ~P A C_ A -> -. { x e. A | x e/ x } e. ~P A )
15		ssrab2 3079	. . 3 |- { x e. A | x e/ x } C_ A
16		elpw2g 3931	. . 3 |- ( A e. V -> ( { x e. A | x e/ x } e. ~P A <-> { x e. A | x e/ x } C_ A ) )
17	15, 16	mpbiri 166	. 2 |- ( A e. V -> { x e. A | x e/ x } e. ~P A )
18	14, 17	nsyl3 588	1 |- ( A e. V -> -. ~P A C_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   e/ wnel 2339   {crab 2352   C_ wss 2973   ~Pcpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-nel 2340  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by:  pwne  3934  pwuninel2  5920